Lindholm, a defenseman, developed his hockey skills with his hometown Skelleftea hockey program in Sweden, starting with the under-16 team. He made his Swedish Hockey League with Skelleftea as an 18-year-old in 2011-12. In 2013-14, he played one regular-season SHL game for Skelleftea and 14 in the playoffs to help them win the SHL championship.
The Colorado Avalanche were impressed enough to select him in the fifth round (No. 144) of the 2014 NHL Draft. Lindholm spent two more seasons with Skelleftea, and signed a three-year entry-level contract with the Avalanche on May 24, 2016.